Autotuning of PID controllers based on relay feedback tests proposed by Astrom and Hagglund in 1981 (respective U.S. Pat. No. 4,549,123 issued in 1985) received a lot of attention after that (W. L. Luyben, “Derivation of Transfer Functions for Highly Nonlinear Distillation Columns”, Ind. Eng. Chem. Res. 26, 1987, pp. 2490–2495; Tore Hagglund, Karl J. Astrom, “Industrial Adaptive Controllers Based on Frequency Response Techniques”, Automatica 27, 1991, pp. 599–609). It identifies the important dynamic information, ultimate gain and ultimate frequency, in a straightforward manner. The success of this type of autotuners lies on the fact that they are simple and reliable. This features of the relay feedback autotuning have lead to a number of commercial autotuners (Tore Hagglund, Karl J. Astrom, “Industrial Adaptive Controllers Based on Frequency Response Techniques”, Automatica 27, 1991, pp. 599–609) and industrial applications (H. S. Papastathopoulou, W. L. Luyben, “Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure”, Ind. Eng. Chem. Res. 29, 1990, pp. 1859–1868).
Luyben (W. L. Luyben, “Derivation of Transfer Functions for Highly Nonlinear Distillation Columns”, Ind. Eng. Chem. Res. 26, 1987, pp. 2490–2495) proposed the use of relay feedback tests for system identification. The ultimate gain and ultimate frequency from the relay feedback test are used to fit a typical transfer function (e.g., first-, second- or third order plus time delay system). This identification was successfully applied to highly nonlinear process, e.g., high purity distillation column. Despite the apparent success of autotune identification, it can lead to signification errors in the ultimate gain and ultimate frequency approximation (e.g., 5–20% error in R. C. Chiang, S. H. Shen, C. C. Yu, “Derivation of Transfer Function from Relay Feedback Systems”, Ind. Eng Chem. Res. 31, 1992, pp. 855–860) for typical transfer functions in process control system. The errors come from the linear approximation (describing function method) to a nonlinear element. The square type of output from the relay is approximated with the principal harmonic from the Fourier series (Derek P. Atherton, “Nonlinear Control Engineering”, Van Nostrand Reinhod: New York, 1982) and the ultimate gain is estimated accordingly. Several attempts were proposed to overcome this inaccuracy but didn't overcome the main source of inaccuracy—linear approximation of the relay element due to the use of describing function method model.